Journal of Economic Intelligence and Mathematics

ABSTRACT

We establish unconditional almost-all ternary results for the restricted weighted Goldbach sum

together with conditional extensions under the Density Hypothesis and the Generalized Riemann Hypothesis.

Main results.

(A) Almost-all via prime anchoring [Proved]. For every $A>0$,

(B) Almost-all with full asymptotic [Proved]. For all odd $n\le X$ outside an exceptional set of size $O_{A,q}\left(X{\left(logX\right)}^{-A}\right)$,

where the ternary singular series $J_{3,a,q}\left(n\right)>0$ is positive and effectively computable. For $(a,q)=(3,4)$ the main term equals $\left(C_{2}S\left(n\right)/4\right)n^2$ with $C_2\in [0.6601618157,0.6601618160]$.

The minor-arc bound $K_{min}(q,A)\le 2.10/\sqrt{\phi \left(q\right)}$ is proved in full, self-containedly, via the exact $L^2$-orthogonality identity $\parallel S_{a,q}{\parallel }_{L^2}^2=\parallel S{\parallel }_{L^2}^2/\phi \left(q\right)$ and a Hölder $(L^2,L^{\infty },L^2)$ factorization. The exceptional-set constant is $C_{ternary}(A,q)\le 4.41/\phi \left(q\right)\cdot 2^A$ [Proved].

Articles 1, 2, and 3 by the same author of this work provide all the black box input data. No GRH, no density hypothesis, and no ternary sieve are used in Parts (A) and (B).

Keywords: Ternary Goldbach problem; Arithmetic progressions; Restricted Goldbach sum; Circle method; Hardy–Littlewood method; Minor arc estimates; Major arc estimates; Dirichlet characters; Exponential sums; Almost-all results; Exceptional set; Singular series; Prime Number Theorem; Vaughan identity; Hölder inequality; L²-orthogonality; Vinogradov–Vaughan bounds; Zero-density hypothesis; Generalized Riemann Hypothesis (GRH); Explicit constants